\chapter{Ultimate performance limitations} % (fold)
\label{cha:toymodel}

The previous chapter presents my results from simulating all-optical switching with transverse optical patterns. The results generated from these simulations  and my experimental results both exhibit increased response time for low switch-beam powers. This chapter presents a conjecture that the increased response time at low switch-beam power is due to critical slowing down. I support this conjecture by mapping the pattern orientation to a simple first-order one-dimensional system. Numerical integration of this model demonstrates an increased response time for weak perturbations, which exhibits a scaling law that is similar to my experimental observations and simulated results.

For clarity, I am using the concept of critical slowing down from the nonlinear dynamics community. In this sense, critical slowing down refers to the situation where the eigenvalue of a system crosses through zero at a bifurcation point leading to a diverging time-scale. This corresponds to the case where a stable fixed point of a system becomes weakly stable and the decay to such a fixed point occurs more slowly.

\section{A toy model} % (fold)
\label{sec:a_toy_model}

A conceptually simple way to describe the dynamics of a system is in terms of a potential that the system seeks to minimize. My experimental observations and numerical simulations suggest that the pattern forming system discussed in this thesis obeys a potential with a ring-shaped well containing six valleys equally spaced around the ring and separated by $\pi/3$ radians (corresponding to 60$^\circ$). The individual valleys represent the six preferred spot locations while the diameter of the well is set by the cone angle $\theta\simeq\sqrt{3\lambda/(\pi L)}$ described in Chapter~\ref{cha:patterns}. A three-dimensional plot of such a potential is shown in Fig.~\ref{fig:toy_v}. It should be noted that this potential is motivated solely by the observed patterns and is thus totally phenomenological and does not correspond to any physical property of the system as far as I know.

\begin{figure}[htbp]
  \begin{center}
    \includegraphics[scale=0.4,viewport=30 80 330 245]{Figures/toy_3d_potential_symmetric.png}
    \includegraphics[scale=0.4]{Figures/toy_3d_potential_symmetric_side.png}
  \end{center}
  \caption[The symmetric potential gives rise to six-spots.]{Two views of the symmetric potential with six wells of equal depth which gives rise to six-spots.}
  \label{fig:toy_v}
\end{figure}

To explore the features of this model in quantitative detail, I examine the functional form of the potential shown in Fig.~\ref{fig:toy_v} given in cylindrical coordinates $\rho$ and $\phi$ by
\begin{equation}
  \label{eqn:v}
  V(\rho,\phi,r,h)=-r e^{-\frac{(\rho - 1)^2}{0.1}} (1 + h \sin{6 \phi}),
\end{equation}
where $r$ and $h$ parameterize various portions of the potential well. The parameter $r$ scales the overall depth of the well and $h$ scales the relative depth of the individual valleys that contribute to the hexagon structure. As an example, for $h=0$, the potential well takes the form of a single ring of uniform depth and for $h=1$ the six wells range from depth $-2r$ to maximum height $V=0$. I have normalized the radial coordinate such that $\rho=d\theta$, where $d$ is the distance from the nonlinear medium to the measurement plane and $\theta$ is the cone opening half-angle of the generated light. There is also an arbitrary constant equal to 0.1 scaling the radial extent of the ring potential, this has been chosen to qualitatively correspond to my observations and does not effect the dynamics of the system along the azimuthal angle.

If I assume that the system finds the global minimum via noise and quantum fluctuations, then the patterns corresponding to the potential shown in Fig.~\ref{fig:toy_v} are expected to exhibit six spots of equal intensity because each well is equally deep. To introduce the possibility of symmetry breaking, I include additional terms in the potential of the form
\begin{equation}
  V_p(\rho,\phi) = (1 + \cos{2 \phi}) \rho^2,
\end{equation}
and correspond to a restoring force that breaks the rotational symmetry such that two of the six spots have slightly lower minima and thus become slightly  preferred. The general effect of this term is to fold the potential upwards in the shape of a taco as shown in Fig.~\ref{fig:taco}. 

\begin{figure}[htbp]
  \begin{center}
    \includegraphics[scale=0.5]{Figures/toy_3d_potential_pump_only.png}
  \end{center}
  \caption[Fundamental symmetry-breaking folds the potential.]{Fundamental symmetry-breaking folds the potential along one axis.}
  \label{fig:taco}
\end{figure}

Introducing two of these symmetry-breaking terms allows me to describe changes in the potential caused by two symmetry-breaking mechanisms: one due to fundamental symmetry-breaking in the system such as via pump-beam misalignment, and the other due to the applied switch beam. The full functional form of the toy-model potential is then given by
\begin{align}
  \label{eqn:full_v}
  V(\rho,\phi,r,h,p,s)&=-r e^{-\frac{(\rho - 1)^2}{0.1}} (1 + h \sin{6 \phi})\nonumber\\
  &+ p \left(1 + \cos\left(2 \left(\phi + \pi/6\right)\right)\right) \rho^2\nonumber\\
  &+ s \left(1 + \cos\left(2 \left(\phi - \pi/6\right)\right)\right) \rho^2,
\end{align}
where the new parameters $p$ and $s$ scale the two symmetry-breaking terms corresponding to the fundamental symmetry-breaking and the switch-beam symmetry-breaking, respectively. Because the switch beam is applied at a fixed azimuthal angle relative to the unperturbed pattern orientation, the only difference between the two new terms are that they are offset by $\Delta\phi=\pi/3$.

To visualize the effect of broken symmetry on the potential, Fig.~\ref{fig:toy_v_pump} shows $V$ for the case where there is only fundamental symmetry breaking ($p\neq0$ and $s=0$). There is clearly a fold in the potential due to the symmetry breaking and from the side view there are now only two global minima rather than six, which can be seen more easily from the side view. These minima corresponding to the two-spot pattern with the orientation preferred by the system.

\begin{figure}[htbp]
  \begin{center}
    \includegraphics[scale=0.4,viewport=30 80 330 245]{Figures/toy_3d_potential_pump.png}
    \includegraphics[scale=0.4]{Figures/toy_3d_potential_pump_side.png}
  \end{center}
  \caption[Fundamental symmetry breaking folds the six-well potential.]{Fundamental symmetry breaking folds the six-well potential and lifts the degeneracy of the wells leaving two spots preferred. The parameters used in Eq.~(\ref{eqn:full_v}) to generate this plot are $p=0.1$, $r=1$, $h=0.1$, and $s=0$.}
  \label{fig:toy_v_pump}
\end{figure}

Alternatively, when the unperturbed system is perfectly symmetric and only the switch beam breaks the symmetry of the system, the potential changes to the one shown in Fig.~\ref{fig:toy_v_switch}. The orientation of the fold in the potential has changed, corresponding to the orientation of the switch beam. From a side view of the potential, there are still two global minima; however, the orientation of these minima has changed by 60$^\circ$ relative to the unperturbed system. The combination of both symmetry-breaking contributions results in a potential that can have either orientation as a global minima depending on the relative strength of the fundamental symmetry-breaking and the switch-beam perturbation.

\begin{figure}[htbp]
  \begin{center}
    \includegraphics[scale=0.4,viewport=30 80 330 245]{Figures/toy_3d_potential_switch.png}
    \includegraphics[scale=0.4]{Figures/toy_3d_potential_switch_side.png}
  \end{center}
  \caption[Potential well for the weakly-perturbed toy model system.]{Potential well for the case of a fundamentally symmetric system that has been perturbed by the switch beam. There are still two global minima but now they have been rotated by $\pi/3$ relative to Fig.~\ref{fig:toy_v_pump}. The parameters used in Eq.~(\ref{eqn:full_v}) to generate this plot are $p=0$, $r=1$, $h=0.1$, and $s=0.1$. }
  \label{fig:toy_v_switch}
\end{figure}

In order to simplify the description of the orientation angle, I move from this two dimensional picture, to a one-dimensional system by considering the circle corresponding to $\rho=1$ and allowing $\phi$ to vary through $2\pi$. This one-dimensional system thus corresponds to a 1D flow along the circle $\rho=1$. The following discussion uses this 1D model to predict the pattern orientation angle in terms of the relative strengths of the two symmetry-breaking terms described above.

\begin{figure}[htbp]
  \begin{center}
    \includegraphics[scale=1]{Figures/toy_potential.pdf}
  \end{center}
  \caption[One-dimensional potential corresponding to a ring.]{The one-dimensional potential corresponding to the ring where $\rho=1$. The curves correspond to $p=0.1$, $r=1$, $h=0.1$, and $s$ between 0 (bottom curve) and 0.5 (top curve) in steps of 0.1. As $s$ increases, the global minimum at $\pi/3$ shifts upward and is replaced by a global minimum at $2\pi/3$ when $s>0.1$. When $s=p=0.1$, the two orientations have equal $V$ and are thus equally preferred.}
  \label{fig:toy_v_1d}
\end{figure}

The change in the 1D potential caused by increasing $s$, which corresponds to increasing the switch beam strength, can be seen in Fig.~\ref{fig:toy_v_1d}. The lower curve corresponds to the potential where there is only fundamental symmetry-breaking. There is a global minimum at $\pi/3$. As $s$ increases, shown by the higher curves, the local minimum at $2\pi/3$ becomes the global minimum as $s$ becomes larger than $p$. Also note, for $s>0.35$, there is only one minimum for $0<\phi<\pi$. Therefore, regardless of the initial orientation of the pattern, applying a switch-beam perturbation corresponding to $s>0.35$ will cause the pattern to rotate. Drawing an analogy between the one-dimensional potential and a ball resting on a hill, we see that, for $s=0$, the orientation state $\phi=\pi/3$ is stable. As $s$ increases the local minimum at $\phi=\pi/3$ becomes shallow, and for $s>0.35$ the minima disappears and the ball rolls down the hill to rest in the nearest stable state $\phi=2\pi/3$.

In order for the ball and hill analogy to be explicitly accurate for this model, it is important to clarify that I consider this system in the overdamped limit. Explicitly, the overdamped limit corresponds to assuming that inertial forces are much smaller than damping forces. From Newton's second law
\begin{equation}
  F=m\ddot\phi+b\dot\phi=-\frac{dV}{d\phi},
\end{equation} 
where I take $m<<b=1$ giving
\begin{equation}
  \dot\phi=-\frac{dV}{d\phi}.
\end{equation}
This simplification also ignores any transients that occur during the pattern formation process and assumes that once the pattern is formed, the dynamics of the pattern orientation $\phi$ is governed by a one-dimensional first-order equation $\dot{\phi}=f(\phi)$. This assumption is not related in any way to the atomic, or optical properties of the system, it is simply made to generate a toy model that serves to demonstrate the topological origin of critical slowing down in a pattern-forming system with weakly broken symmetry. A more accurate analogy for Fig.~\ref{fig:toy_v_1d}, to borrow from Strogatz \cite{Strogatz_2001aa}, would be of a ball rolling down a hill through a layer of viscous goo so there is no possibility of overshoot or oscillation.

The temporal dynamics of this one-dimensional system can be observed via numerical integration of the first-order equation
\begin{equation}
  \dot{\phi}=-\frac{dV}{d\phi}=-6 h r \sin 6 \phi + 2 p \sin\left(2\left(\phi+\pi/6\right)\right) + 
   2 s \sin\left(2\left(\phi - \pi/6\right)\right),
\end{equation}
where the parameters $r$, $h$, $p$, and $s$ are as before. The depth of the potential well is scaled by $r$ so for the remainder of this chapter, I set $r=1$. Similarly, the preference toward hexagon patterns is parameterized by $h$ where $h=1$ yields six potential wells of depth $r$ and $h=0$ leaves the entire 1D system at a constant potential. For the purposes of illustrating this toy model, I choose $h=0.1$ so the height of the barrier between each of the six spots is one tenth of the depth of the ring-shaped well. The relative strengths of $s$ and $p$ are of primary interest to the problem so I set $p=0.1$ and allow $s$ to vary over the range $[0,0.5)$.

Once the applied symmetry-breaking perturbation overcomes the fundamental asymmetry of the system, the orientation $\phi$ will change. This corresponds to the switch action I observe experimentally: with an applied switch-beam, I can change the orientation of the generated patterns. The toy model switch response is shown in Fig.~\ref{fig:toy_resp} for increasing values of $s$, analogous to increasing amounts of switch-beam power. The first curve shows that for $s=0.35$, the initial state is stable even after the perturbation is applied. In the language of nonlinear dynamics, the point $\phi_0=\pi/3$ is a weakly-stable fixed point of the system. For $s>0.35$ the pattern angle $\phi$ changes from $\pi/3$ to $2\pi/3$ corresponding to rotation by 60$^\circ$.

\begin{figure}[htbp]
  \begin{center}
    \includegraphics[scale=1]{Figures/toy_response.pdf}
  \end{center}
  \caption[Toy model response to perturbations of various strength.]{The 1D model response to symmetry-breaking perturbations in the range $0.35\leq s<0.49$. Once the applied perturbation overcomes the potential barrier between two orientation states, the orientation changes. This change occurs more quickly for larger perturbations (\emph{i.e.}, for larger $s$).}
  \label{fig:toy_resp}
\end{figure}

To compare the behavior of this model to my experimental and numerical results, I measure the response time $\tau_\mathrm{1D}$ as the time between the initial perturbation (in the case of the toy model $t=0$) and the time $\phi$ passes a threshold set at $\phi_\mathrm{on}=1.5$. The measured response time as a function of the perturbation parameter $s$ is shown in Fig.~\ref{fig:toy_rt}. The response time is slowest ($\tau_\mathrm{1D}$ is largest) for perturbations that are just above the threshold for switching, which is consistent with my experimental observations and  simulation results. This phenomena can be explained by critical slowing down, which is known to occur near weakly-stable fixed points in a one-dimensional flow \cite{Strogatz_2001aa}.

\begin{figure}[htbp]
  \begin{center}
    \includegraphics[scale=1]{Figures/toy_resp_times.pdf}
  \end{center}
  \caption[The response time measured for the toy model.]{The response time measured for the 1D model shows critical slowing down as the perturbation parameter $s$ decreases towards the critical value $s_c=0.35$. For $s<0.35$, no pattern rotation occurs.}
  \label{fig:toy_rt}
\end{figure}

The response time $\tau_\mathrm{1D}(s)$ increases as a power law as $s$ decreases toward the critical value for pattern rotation $s_c=0.35$. A fit of this functional form yields
\begin{equation}
  \tau_\mathrm{1D}(s)\propto(s-s_c)^{-0.8 \pm 0.02},
\end{equation}
where the critical value $s_c=0.35$ is the maximum value of $s$ where $\phi$ is stable, \emph{i.e.}, where the pattern does not rotate because it is pinned by the fundamental symmetry breaking.

% section a_toy_model (end)

\section{Performance implications} % (fold)
\label{sec:performance_implications}

Interesting implications arise for the ultimate performance of my switch after considering this simple 1D model. My experimental results, simulations of an analogous nonlinear optical system, and this model based on symmetry arguments all exhibit an increase in the response time as the strength of the applied perturbation decreases. In the case of my experiments and simulations, this perturbation is the injected switch beam. The 1D model suggests that the strength of the applied perturbation must be sufficient to overcome fundamental symmetry-breaking that exists in the unperturbed system. The functional relationship between the perturbation strength and the response time is given as an inverse power law that diverges at the critical perturbation strength. If the perturbation strength is below this critical value, no pattern rotation occurs.

\begin{figure}[htbp]
  \begin{center}
    \includegraphics[scale=1]{Figures/sim_power_law.pdf}
  \end{center}
  \caption[Simulated response time increases as a power law.]{The response time of the simulated switch follows a power law $\tau_\mathrm{sim}(P_s)\propto P_s^{-0.3 \pm 0.01}$.}
  \label{fig:sim_power_law}
\end{figure}

For comparison, the results from numerical simulation of my pattern-based switch also follow an inverse power law. However, the response time $\tau_\mathrm{sim}$ diverges at a critical value corresponding to zero switch-beam power (see Fig.~\ref{fig:sim_power_law}). This implies that any applied perturbation will cause the pattern to rotate, although weak perturbations cause the rotation to be extremely slow. A critical value of zero perturbation strength is consistent with the above interpretation considering that there is no fundamental symmetry-breaking introduced in the numerical simulation so the patterns have no universally-preferred orientation. My simulations begin by seeding the pattern in a specific orientation, and then causing it to rotate to a new orientation with a perturbation. This is different from biasing the pattern orientation by introducing fundamental symmetry breaking.

In the simulated switch, critical slowing down causes the pattern to rotate more slowly for weak perturbations. No orientation is preferred over another unless the switch beam is applied, thus the response time diverges at zero switch-beam power rather than at some finite critical value. The exponent found in the power law fit for $\tau_\mathrm{sim}$ is $-0.3 \pm 0.01$, which is closer to agreement with the 1D toy model if the parameter $s$ is taken to be the field strength. Assuming this relationship, the power law exhibited by the 1D model ($\tau_\mathrm{sim}\propto s^{-0.8 \pm 0.02}$) becomes $\tau_\mathrm{sim}\propto s^{-0.4 \pm 0.01}$ due to the quadratic relationship between power and field strength.

\begin{figure}[htbp]
  \begin{center}
    \includegraphics[scale=1]{Figures/exp_power_law.pdf}
  \end{center}
  \caption[Experimental response increases as a power law.]{The response time measured experimentally follows a power law $\tau_\mathrm{exp}(P_s)\propto (P_s-P_c)^{-0.22 \pm 0.01}$ where $P_c=4.3\pm0.3\times10^{-8}P_p$. The vertical dashed line indicates $P_c$.}
  \label{fig:exp_power_law}
\end{figure}

The experimental data exhibits similarities to the simulation and the 1D model, although the range of switch-beam powers over which I can collect data is limited by the fact that the patterns exhibit partial switching for low switch-beam powers. The amplitude of the switch response decreases at low switch-beam powers which prevents measurement of the response time in the limit of zero switch-beam power. Thus, one weakness of the toy model is that it does not predict the transistor-like behavior that I observe in my experiment. As discussed in Chapter~\ref{cha:nummodel}, the transistor-like response is also not observed in simulations based on a Kerr medium. For this reason, it is likely that the absorption in the experimental system is responsible for this behavior.

The available response time data, presented first in Chapter~\ref{cha:switch}, spans less than an order of magnitude, so an inverse power-law fit is not entirely conclusive. However, the data is consistent with a critical switch-beam power of less than 35~pW. An approximate fit, shown in Fig.~\ref{fig:exp_power_law}, corresponds to a critical switch-beam power of $P_c=4.3\pm0.3\times10^{-8}P_p=24\pm2$~pW. The lowest measured switch response was for $P_s=35$~pW, which corresponds to 600~photons for $\tau_\mathrm{exp}=5\,\mu$s. If the experimental response time does diverge at $P_c=24\pm2$~pW, then the lowest possible switching photon number must be between 600$\pm40$~photons and $\tau_\mathrm{exp} P_c/E_p=470\pm40$~photons.

% Clearly the partial switch response limits the performance along with critical slowing down. Neither the 1D toy model, nor the simulated switch exhibit partial switching, so one possible conclusion is that partial switching is due to the presence of absorption. In this case, exploring pattern-forming systems without significant absorption may be promising for future work. If absorption is eliminated can account for partial switching but the pattern-forming mechanism in the transparent Kerr model is not limited by CSD.

% Absorption has not been considered in my numerical simulation, nor in this toy model. Although most of the phenomena I observe are consistent with theoretical treatments that ignore absorption, the observation of partial switching is not. 

% section performance_implications (end)

\section{Summary} % (fold)
\label{sec:toy_summary}

% section summary (end)

Critical slowing down appears to be fundamental to my pattern-based switch. Using symmetry arguments, the toy model presented in this chapter shows the response time follows an inverse power law that diverges at a critical value of the symmetry-breaking perturbation. This suggests that the primary limitation on the switch sensitivity is likely due to additional factors such as absorption and not due to fundamental symmetry-breaking or pinned patterns. If the patterns were pinned by fundamental symmetry-breaking, I would expect to observe a power-law increase in the response time that diverges at a finite switch-beam power. Instead, my data shows that, if a critical switch-beam power exists, it is less than 35~pW and indicates the minimum number of photons capable of actuating my all-optical switch is between 470 and 600 photons.


% chapter ultimate_performance_limitations (end)